Article
Ufa Mathematical Journal
Volume 10, Number 4, pp. 40-49
On uniqueness of weak solution to mixed problem for integro-differential aggregation equation
Vil'danova V.F.
DOI:10.13108/2018-10-4-40
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In a well-known paper by A.Bertozzi, D.Slepcev (2010), there was established
the existence and uniqueness of solution to a mixed problem for
the aggregation equation
ut−ΔA(x,u)+div(u∇K∗u)=0
describing the evolution of a colony of bacteria in a bounded convex domain Ω. In this paper we prove the existence and uniqueness of
the solution to a mixed problem for a more general equation
β(x,u)t=div(∇A(x,u)−β(x,u)G(u))+f(x,u).
The term f(x,u) in
the equation models the processes of ``birth-destruction'' of
bacteria. The class of integral operators G(v) is wide enough
and contains, in particular, the convolution operators ∇K∗u. The vector kernel g(x,y) of the operator G(u) can have singularities.
Proof of the uniqueness of the solution in the work by A.Bertozzi, D.Slepcev was based on the conservation of the mass ∫Ωu(x,t)dx=const of bacteria and employed the convexity of Ω and the properties of the convolution operator. The presence
of the ``inhomogeneity'' f(x,u) violates the mass conservation. The proof of
uniqueness proposed in the paper is suitable for a nonuniform
equation and does not use the convexity of Ω.